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\title{Probing the Halo Environments of Globular Clusters with Cool White Dwarfs}
\author{Travis J. Hurst, Andrew R. Zentner, Aravind Natarajan and Carles Badenes}
\affiliation{Department of Physics and Astronomy, \& The PITTsburgh Particle Physics and Cosmology Center (PITT PACC),University of Pittsburgh, Pittsburgh, PA 15260, USA}
\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
White dwarfs capture dark matter as they orbit their host halos. If the dark matter is self-interacting, then these captured particles will annihilate, heating the core and preventing the white dwarf from cooling. Because the annihilation rate is directly proportional to the local dark matter density, observations of cool white dwarfs can be used to constrain their dark matter environments. In particular, if the parameters of the dark matter particle are known, then the temperature of the coolest white dwarf in a globular cluster can be used to constrain the dark matter density of the cluster's halo (potentially even ruling out the presence of a halo if the inferred density is of order the ambient Galactic density). Recently several direct detection experiments have seen signals whose origins are perhaps due to low mass dark matter. In this paper, we show that if these claims from CRESST, DAMA, CDMS-Si, and CoGeNT are correct, then observations of NGC 6397 limit the fraction of dark matter in that cluster to be $\lesssim 10^{-3}$.  This is an improvement over the existing constraint by 3 orders of magnitude.
\end{abstract}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
In the current cosmological paradigm, $\approx 23\%$ of the energy of the Universe is in the form of an unidentified form of Dark Matter (DM) \cite{WMAP9,Planck}. The nature of this DM is currently unknown, but the most popular particle candidates are Weakly-Interacting Massive Particles (WIMPs). WIMPs are compelling candidates because, among other reasons, WIMPs can be produced thermally in the early Universe such that their relic abundance yields the correct value for the DM density, $\Omega_{\mathrm{DM}} \approx 0.23$. Moreover, WIMPs arise naturally in various extensions to the Standard Model (SM) of particle physics. As such, detecting, identifying, or excluding WIMP-like DM is a high priority in cosmology and particle physics. Accordingly, a great deal of resources have been brought to bear on this problem.

One avenue of inquiry into the nature of DM is direct detection. Direct detection experiments seek to identify DM by observing scattering events in a detector on Earth. A number of direct detection experiments have reported intriguing results recently including Coherent Germanium Neutrino Technology (CoGeNT) \cite{CoGent}, the silicon detectors of the Cryogenic Dark Matter Search (CDMS-Si) \cite{CDMS-Si}, Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) \cite{CRESST} and Dark Matter Large sodium Iodide Bulk for RAre processes (DAMA/LIBRA, henceforth just DAMA) \cite{DAMA}. These experiments all see events that are inconsistent with expected backgrounds. Taking all of these experiments together, there is no consistent interpretation of these events as WIMP scattering (though there are some models that can alleviate the tension between the experiments e.g. refs. \cite{Kelso,Foot,Zurek,Panci1,Panci2}). 

Broadly speaking these data, interpreted as DM, point toward a DM particle with a mass of $M_\chi \sim$ 5-25 GeV and cross-section for collision off a proton $\sigma_{\chi\mathrm{p}} \sim 10^{-41}$ cm$^2$ (see fig. \ref{fig:mvsigma}). Recent results from the Large Underground Xenon Dark Matter (LUX) \cite{LUX} and SuperCDMS \cite{SuperCDMS} experiments exclude nearly all of the parameter space consistent with a low mass DM WIMP interpretation of these other experiments, casting serious doubt on the viability of interpreting these events as due to DM.  E.g. ref. \cite{Zurek} found that even when considering non-standard models of WIMP DM scattering such as an Anapole interaction, which can bring the regions of interest for CDMS-Si, CoGeNT, and DAMA into alignment, the parameter space preferred by these experiments is excluded by the LUX results.

A second avenue of inquiry into the nature of DM is indirect detection.  Indirect detection experiments seek to identify the DM by observing the particles (and their subsequent decay products) produced by the annihilation of self-interacting DM in astrophysical sources such as the Sun \cite{Modak}.  

Several groups have found strong evidence that there is an extended gamma ray emmision coming from the Galactic Center using data from the Large Area Telescope aboard the Fermi Gamma Ray Space Telescope (Fermi-LAT).  This emission is consistent with the annihilation of a low mass ($\sim$ 10-30 GeV) thermal WIMP DM particle to quarks \cite{Abazajian, Good&Hooper, Hooper&Linden, Abaz&Kap, Gordon&Mac, Mac&Gordon} or leptons \cite{Abazajian, Hooper&Good, Hooper&Linden, Abaz&Kap, Gordon&Mac, Mac&Gordon, Kyae}, but the evidence is not conclusive (e.g. \cite{Boyarsky}).
 
Successful detection of DM opens the door to what some authors have referred to as WIMP astronomy (e.g. ref. \cite{Peter}). The premise of WIMP astronomy is that with the properties of DM known, observations of astronomical phenomena such as cosmic ray fluxes or stellar evolution can yield further information about the structure and substructure of the Milky Way's Galactic DM halo. A potential application of WIMP astronomy is the following: It is well known that a star should capture DM as it orbits its host halo (see e.g. the review \cite{Jungman} and references therein). Therefore, if DM is self-annihilating there should be some heating from DM annihilation in the cores of stars \cite{Burners,Burners2}. The effects of such annihilations have been considered for Main Sequence (MS)  stars \cite{Burners3}, primordial stars \cite{Spolyar, Natarajan, Iocco} and White Dwarfs (WDs) \cite{Hooper}.   
%
%	0.6 M_solar WD cooling curve
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{WDcool}
\caption{The cooling curve of a 0.6 M$_\odot$ WD modeled with \texttt{MESA star}: Note that in order for the WD to reach temperatures of order a few $\times\ 10^3$ K, the cooling time must be of order the age of the Galaxy.}
\label{fig:WDcool}
\end{figure}
%
%

If WDs are heated by DM annihilations in their cores, then they will be prevented from cooling below some minimum temperature \cite{Hooper}. Hence, given a DM model, we can put a maximal constraint on the local DM density around a distant WD by assuming all of its luminosity is from DM annihilation. Alternatively, given information on the DM environment of the WD from kinematic observations, we could constrain the DM model (note that this would require an observed {\it lower bound} on the amount of DM).

In the present work, we explore the potential for observations of cool WDs in Globular Clusters (GCs) to place limits on the DM content of the clusters. We then consider NGC 6397 as a specific example, because it has been observed extensively with the Hubble Space Telescope (HST) \cite{Heyl,Stello,DeMarchi,Hansen,Richer}. We will present results for WIMP-like DM candidates with properties that may explain the anomalous events in the DAMA, CoGeNT, CRESST, and CDMS-Si detectors as well as WIMP-like DM that is consistent with the recent LUX and SuperCDMS limits.

GCs are interesting in the context of WIMP astronomy, because they are the largest structures known that show no evidence of DM. Moreover, viable formation and evolution scenarios for the Galactic GCs remains controversial, partly because of the complex element abundance patterns measured among the stars in several GCs \cite{Gratton,Gratton2012,Con&Sperg}. As an example of the state-of-the-art constraints on DM in GCs, ref. \cite{Conroy} found that for NGC 2419 and MGC1 $M_{\mathrm{DM}}/M_* \lesssim 1$ where $M_{\mathrm{DM}}$ is the mass in DM of the cluster and $M_*$ is the stellar mass of the cluster. Subsequently ref. \cite{Shin} found a similar result for NGC 6397. 

In the present work we show how WIMP astronomy may improve upon these types of constraints by several orders of magnitude.  In particular, we find that for a DM WIMP with $M_\chi \sim 10 \ \mathrm{GeV} \ \mathrm{and} \ \sigma_{\chi\mathrm{p}} \sim 10^{-41} \ \mathrm{cm}^2$ the density of DM in NGC 6397 must be a factor of $\sim 10^{3}$ times less than the average density.  

The remainder of the paper is organized as follows:  In \S II we show how the local DM density can be related to the luminosity of the WD once the WD has reached it's minimum temperature.  In \S III we consider the particular case of NGC 6397 and present our derived constraint on the fraction of DM in this GC.  In \S IV we summarize our results and discuss the potential for future applications of our method.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methods}

To calculate the rate at which a WD will capture DM we use the result (A.16) from \cite{Zentner}
%
%	Definition of the capture rate      
\begin{equation}
\label{eq:capture_rate}
	C_{\mathrm{c}} = \sqrt {\frac{3}{2}}\frac {\rho_\chi}{M_\chi}\sigma_i v_{\mathrm{esc}}(R)\frac{v_{\mathrm{esc}}(R)}{\bar{v}}N_i \langle\hat{\phi}\rangle\frac{\mathrm{erf}(\eta)}{\eta},
\end{equation}
%
%
where $\rho_\chi$ is the local DM density, $R$ is the radius of the WD, $v_{\mathrm{esc}}(R)$ is the escape speed at the surface of the star, $\bar{v}$ is the dispersion of the DM velocity profile (assumed to be Maxwell-Boltzmann), $\eta$ is the ratio of the star's velocity through its halo to the local velocity dispersion of the halo (assumed to be of order unity) and 
%
%	Phi hat
\begin{equation}
	\hat{\phi} = \frac{v_\mathrm{esc}^2(r)}{v_\mathrm{esc}^2(R)}
\end{equation}
%
%
is a dimensionless potential for stellar nucleons.  $\langle\hat{\phi}\rangle \approx$ 2.4 denotes the average of $\hat{\phi}$ over all nucleons in the star.  The cross-section for scattering off of nuclear species $i$ is $\sigma_i$, $N_i$ is the number of nucleons of species $i$ in the star, and the total capture rate is the sum over all species. Note that we are justified in using eq. (\ref{eq:capture_rate}), because for the WIMPs we consider, any energy loss due to scattering results in the WIMP being captured.

Scattering off a given nucleus is coherent at the relevant energies, so the cross-section for scattering off a given nucleus is approximately 
%
%	Approximate scattering x-section      
\begin{equation}
	\sigma_{\mathrm{N}} \approx \sigma_{\chi\mathrm{p}}A^2\frac{M_\chi^2M_{\mathrm{N}}^2}{{(M_\chi+M_{\mathrm{N}})}^2}\frac{{(M_\chi+m_{\mathrm{p}})}^2}{M_\chi^2m_{\mathrm{p}}^2},
\end{equation}
%
%
where $A$ is the atomic mass number of the nucleus of interest, $M_{\mathrm{N}}$ is the mass of the nucleus, and $m_{\mathrm{p}}$ is the proton mass.  For simplicity we will consider a WD made entirely from Carbon and Oxygen.  Then we can write the composition of the WD as
%
%	WD composition   
\begin{equation}
	N = N_{\mathrm{C}}  + N_{\mathrm{O}} = f_{\mathrm{C}}\frac{M_{\mathrm{WD}}}{M_{\mathrm{C}}} + f_{\mathrm{O}}\frac{M_{\mathrm{WD}}}{M_{\mathrm{O}}},
\end{equation}
%
%
where $f_{\mathrm{C}}$ and $f_{\mathrm{O}}$ are the fractions of Carbon and Oxygen and obviously $f_{\mathrm{C}} + f_{\mathrm{O}} = 1$.  (Our results are not sensitive to the WD's exact composition).  
%
%	WD cooling sequence plot
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{WDplot}
\caption{The WD cooling sequence of NGC 6397 \cite{Hansen,Richer}:  The important feature is the sharp decline in the number of WDs with apparent magnitudes greater than F814W = 27.6, indicating that there is a real truncation of the WD cooling sequence.  The Main Sequence can be seen in the upper-right corner.  The data points were obtained from ref. \cite{CDS}.}
\label{fig:WDplot}
\end{figure}
%
% 

We consider stable, WIMP-like (annihilation rate $\sim 3 \times 10^{-26}\ \mathrm{cm}^3$/s) DM of mass $M_\chi \gtrsim$ 70 keV.  WIMPs with  $M_\chi \lesssim$ 70 keV can escape the star via evaporation---the ejection of WIMPs by hard elastic scattering from nuclei.  We can estimate the evaporation mass by demanding that the typical velocity of a WIMP $v \sim (T_\mathrm{c}/M_\chi)^{1/2}$ be less than the local escape speed of the star $\sim 1.1 \times 10^4$ km/s.  Here $T_\mathrm{c}$ is the temperature in the core of the star \cite{Jungman}.  Below we will consider a WD with $M_\mathrm{WD} \approx 0.6 \   \mathrm{M}_\odot \ \mathrm{and} \ T_\mathrm{eff} \approx$ 3700 K.  Using the publicly available stellar evolution code \texttt{MESA star} \cite{MESA1,MESA2,MESAsite} we find that such a WD should have a core temperature $T_\mathrm{c} \sim 10^6$ K corresponding to an evaporation mass $\sim$ 70 keV.  (For a more careful discussion of the evaporation mass see \cite{Griest&Seckel}).

The number of WIMPs captured within the star, $N_\chi$, is then governed by the differential equation 
%
%	Diff eq for N_chi     
\begin{equation}  
	\frac{\mathrm{d}N_\chi}{\mathrm{d}t} = C_{\mathrm{c}} - C_{\mathrm{a}}N_\chi^2,
\end{equation}
%
%
where $C_{\mathrm{a}}$ is twice the rate of annihilation events (because each annihilation eliminates 2 particles). The solution to this equation for homogeneous initial conditions is
%
%	Solution for N_chi    
\begin{equation}
	N_\chi = \sqrt{\frac{C_{\mathrm{c}}}{C_{\mathrm{a}}}}\tanh\left(\sqrt{C_{\mathrm{c}}C_{\mathrm{a}}}t\right).
\end{equation}
%
%

There is a timescale for equilibration between DM annihilation and capture, $\tau_{\mathrm{eq}} = 1/\sqrt{C_{\mathrm{c}}C_{\mathrm{a}}}$, such that for $t > \tau_{\mathrm{eq}}$, $N_\chi$ approaches a steady state solution $N_{\chi,\mathrm{eq}} = \sqrt{C_{\mathrm{c}}/C_{\mathrm{a}}}$ \cite{Zentner}.  The annihilation rate at equilibrium within a WD will be
%
%	Annihilation rate
\begin{equation}
	\Gamma_{\mathrm{a}} = \frac{1}{2}C_{\mathrm{a}}N_{\chi,\mathrm{eq}}^2 = \frac{1}{2}C_{\mathrm{c}},
\end{equation}
%
%
because there are $N_{\chi,\mathrm{eq}}^2$/2 distint pairs of DM particles within the star.  To calculate $\tau_{\mathrm{eq}}$ we first express $C_{\mathrm{a}}$ in terms of effective volumes
%
%	Eqn for C_a
\begin{equation}
	C_{\mathrm{a}} = \langle\sigma_{\mathrm{A}}v\rangle\frac{V_2}{V_1^2},
\end{equation}
%
%
where $\langle\sigma_{\mathrm{A}}v\rangle$ is the thermally averaged annihilation cross-section, and 
%
%	Effective volume
\begin{equation}
	V_j = 3.31\times 10^{19}\left(\frac{100\ \mathrm{GeV}}{jM_\chi}\right)^{3/2}\mathrm{cm}^3
\label{eq:v_eff}
\end{equation}
%
%
is the effective volume of captured WIMPs within the star for $j$ = 1 \cite{Griest&Seckel,Zentner}.  

Note that it is assumed in eq. (\ref{eq:v_eff}) that over the volume of interest (the core of the WD) the density is $\rho(r) = \rho_\mathrm{c} = 4 \times 10^6$ g/cm$^3$ and the temperature is $T(r) = T_\mathrm{c} = 10^6 \ \mathrm{K}$ where the subscript c denotes the value in the core and both $\rho_\mathrm{c}$ and $T_\mathrm{c}$ were found with \texttt{MESA star}.  

For a thermal WIMP with $\langle\sigma_{\mathrm{A}}v\rangle = 3 \times 10^{-26} \ \mathrm{cm}^3/\mathrm{s}$, $\sigma_{\chi\mathrm{p}} = 10^{-41}\ \mathrm{cm}^2$ and mass $M_\chi = 10$ GeV, captured by a WD in the mass range we consider and located in NGC 6397, we find that $\tau_{\mathrm{eq}} = 2.5 \times 10^3$ years for the fiducial constraint $\rho_\chi = \rho_*$.  While for our more restrictive constraint $\rho_\chi \lesssim \rho_*\times 10^{-3}$ (dervied below) we find that $\tau_{\mathrm{eq}} \gtrsim 8 \times 10^4$ years.  Note that in either case this is many orders of magnitude less than the age of the cluster $\sim 10$ Gyr.  In fact, we find that $\tau_{\mathrm{eq}}$ is of order the age of the cluster only for $\langle\sigma_{\mathrm{A}}v\rangle \lesssim 10^{-39}\ \mathrm{cm}^3/\mathrm{s}$ for the fiducial constraint and $\langle\sigma_{\mathrm{A}}v\rangle \lesssim 10^{-36}\ \mathrm{cm}^3/\mathrm{s}$ for our more restrictive constraint.  Hence, for a thermal WIMP-like ($\langle\sigma_{\mathrm{A}}v\rangle \sim 3 \times 10^{-26} \ \mathrm{cm}^3/\mathrm{s}$) DM particle, we are justified in using the equilibrium result $N_{\chi ,\mathrm{eq}}$.
%
%	Neutrno opacity figure
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{MFP}
\caption{The mean-free path for neutrinos in a 0.6 M$_\odot$ WD:  The horizontal dotted line is the radius of the WD.  When $M_\chi \sim$ a few MeV, the WD will not be opaque to neutrinos, because the MFP is of order the WD radius.  Hence, we shall only consider values of $M_\chi \gtrsim$ a few MeV.}  
\label{fig:MFP}
\end{figure}
%
%

Due to the accumulation of DM within WDs, these stellar remnants can be used to probe the 
physics of DM and the DM content of a variety of astrophysical environments. 
As in ref. \cite{Hooper}, if DM annihilations occur, they would produce a luminosity 
%
%	DM luminosity
\begin{equation}
	L_\chi \approx \Gamma_{\mathrm{a}}M_\chi.
\end{equation}
%
%
Notice that nearly all of the energy released at the annihilation will be deposited in the interior of the WD so long as $M_\chi \gtrsim$ a few MeV.  To see this, first note that the total elastic scattering cross-section for a neutrino and a proton is \cite{cupp}
%
%	Neutrino-proton cross-section
\begin{equation}
	\sigma_{\nu\mathrm{p}} \approx 6.0 \times 10^{-40}\ \mathrm{cm}^2\left(\frac{E_\nu}{1\ \mathrm{GeV}}\right)^2.
\end{equation}
%
%
The mean-free path (MFP) is 
%
%	mean-free path
\begin{equation}
	l = (n\sigma)^{-1}
\end{equation}
%
%
where $n \approx 5.2\ \times 10^{35}\ \mathrm{protons}/\mathrm{cm}^3$ is the projected surface density of the WD (calculated with \texttt{MESA star}).  Fig. \ref{fig:MFP} compares the MFP to the radius of the WD for $E_\nu = M_\chi,\ 0.1\ M_\chi\ \mathrm{and}\ 0.01\ M_\chi$. We see that our results shall be robust so long as $M_\chi \gtrsim$ a few MeV.  At lower masses, the WD will not be opaque to neutrinos regardless of the portion of the energy from annihilation received by the neutrino. However, as the only SM particle below this mass is the electron, there are few decay channels at such low DM masses.  Therefore, our method should be applicable to most self-interacting DM models.  

Furthermore, notice that the capture rate varies in inverse proportion to $M_\chi$ so long as the DM mass is not very different from the mass of the target nucleus (see eq. (\ref{eq:capture_rate}) and ref.~\cite{Zentner}).  Thus, the annihilation luminosity is nearly independent of mass over a few orders of magnitude of DM particle mass centered around $M_\chi \sim$ 10-20 GeV.  The total luminosity of the WD can be related to its structural properties by 
%
%	WD luminosity
\begin{equation}
	L_{\mathrm{WD}} = 4\pi R^2\sigma_{\mathrm{SB}}T_{\mathrm{eff}}^4,
\end{equation}
%
%
where $\sigma_{\mathrm{SB}}$ is the Stefan-Boltzmann constant, $R$ is the radius of the WD, 
and $T_{\mathrm{eff}}$ is the effective temperature of the WD. 

WDs cool and dim over time (see figs. \ref{fig:WDcool} \& \ref{fig:WDplot} and ref. \cite{HansenReview}).  If DM annihilations occur at a significant rate within a WD, then annihilations constitute a source of energy within the WD that can eventually provide an amount of energy comparable to the luminosity of the WD. If this occurs, the WD cooling will be halted at some minimum luminosity, $L \approx L_\chi$ from above, and minimum effective surface temperature. Consequently, The observed luminosities and surface temperatures of WDs can be used to place bounds on DM WIMP properties and the distribution of DM WIMPs local to the WDs. (Though we do not discuss them here, another well motivated DM particle candidate is the axion.  It is worth noting that WDs are also used to constrain the parameters of axions, as they would also have observable effects on WD cooling \cite{Raffelt}.)
%
%	rho vs. sigma figure
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{rhovsigma}
\caption{The DM fraction vs. DM-proton scattering cross-section for NGC 6397:  The shaded band represents the constraints for the best fit values of $M_\chi$ from the direct detection experiments and spans the range 3500-3700 K corresponding to the temperature of the WD at the truncation of the cooling sequence.  (The constraints from CoGeNT, CRESST, DAMA and CDMS all lie within this band). The horizontal dotted lines show the fiducial contraints, $8.5 \times 10^{-6} < {\mathrm{f_{DM}}} < 0.5$. The solid vertical lines denote the best fit cross-sections for CoGeNT, CRESST, DAMA and CDMS as indicated by the labels on the plot.  The vertical dashed lines indicate the exclusion limits from LUX and SuperCDMS at several different values of $M_\chi$ as indicated by the labels on the plot.  Note that if any of the scattering events from CoGeNT, CRESST, DAMA and CDMS are confirmed as DM WIMPs, then the fraction of DM in NGC 6397 must be at least 3 orders of magnitude below the fiducial upper limit, and is within 2 orders of magnitude of the ambient Galactic DM density.}  
\label{fig:rhovsigma}
\end{figure}
%
%

This is the effect that we aim to exploit. In a particular astrophysical environment (we will explore GCs) WIMP-like DM particles will annihilate within WDs providing a source of annihilation luminosity that is proportional to the DM-nucleon scattering cross-section $\sigma_{\chi \mathrm{p}}$ and the local DM density $\rho_\chi$, and inversely proportional to the local DM velocity dispersion $\bar{v}$. Typical DM velocities must be similar to the local stellar velocities for an equilibrium structure, so $\bar{v}$ is informed by observational data and has little parametric freedom compared to the DM density and cross-section. According to this argument, the minimum luminosity of WDs in a given environment constrains the product $\rho_\chi \sigma_{\chi \mathrm{p}}$ such that $L \gtrsim L_\chi$. In the following section, we will present constraints on the DM density and scattering cross-section based on high-quality observations of a nearby GC.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}

NGC 6397 is a metal-poor GC located at a distance of 2.6 kpc from the Sun (making it one of the 2 closest GCs along with M4) \cite{Harris,Reid}.  This cluster has a mass $M_{6397} = 1.1\ \pm\ 0.1 \times 10^5$ M$_\odot$ \cite{Heyl}, half-light radius $R_{\mathrm{HL}}$ = 2.2 pc, and is one of 29 Galactic GCs which has undergone a core-collapse \cite{Harris}.  Most importantly for the present work, the WD cooling sequence of NGC 6397 has been measured to unprecedented depth and precision by the Advanced Camera for Surveys (ACS) aboard HST \cite{Hansen,Richer} (fig. \ref{fig:WDplot}), making it an ideal candidate for testing our method.

We can estimate the average density of NGC 6397 as
%
%	Average Density
\begin{equation}
	\bar{\rho} \approx \frac{M_{6397}/2}{\frac{4}{3}\pi R_{\mathrm{HL}}^3} \approx 4.7 \times 10^4\ \mathrm{GeV/cm}^3
\end{equation}
%
%
where we have assumed that half of the GC mass is enclosed by the half-light radius.  Note that the mass measurement comes from the dynamics of the cluster; therefore, this average density includes any DM halo that may be present.  As current bounds on the amount of DM are roughly at the level of $M_{\mathrm{DM}}/M_{\mathrm{GC}} \lesssim 1$, and there is very little guidance on the structures of DM halos that GCs may have had early in their evolution, we parameterize the amount of DM in the cluster by the fraction of DM
%
%	Definition of f_DM
\begin{equation}
	f_{\mathrm{DM}} = \rho_\chi/\bar{\rho}.
\end{equation}
%
%
The fiducial constraints on $f_{\mathrm{DM}}$ are then $8.5\times 10^{-6} \lesssim f_{\mathrm{DM}} \lesssim 0.5$ where the upper limit is from ref. \cite{Shin} and the lower limit is the ambient density of the Milky Way's DM halo---assumed to follow a Navarro-Frenk-White (NFW) profile \cite{NFW}.  In calculating this lower limit we used the favored halo model of ref. \cite{Klypin} and a galactocentric distance for NGC 6397 of 6.0 kpc \cite{Harris}.
%
%	rho vs. mass figure
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{rhovmass}
\caption{The DM fraction vs. DM particle mass for NGC 6397:  The colored bands are the constraints for the best fit values of $\sigma_{\chi\mathrm{p}}$ from CoGeNT, CRESST, DAMA and CDMS and span the range 3500-3700 K corresponding to the temperature of the WD at the truncation of the cooling sequence in NGC 6397.  The vertical dashed lines denote the best fit mass for the corresponding experiment.  Also shown are the 90\% exlusion limits from LUX and SuperCDMS \cite{DMTools}.}
\label{fig:rhovmass}
\end{figure}
%
%

The density and surface brightness profiles of GCs are typically modelled with King models which have constant density cores \cite{King1962,King1966}.  (As a post core-collapse cluster, NGC 6397 has a more centrally concentrated profile than the typical King model).  Note, that this is in contrast to the density profile of DM halos, which increase like $1/r$ towards their centers \cite{NFW}.  Thus, our parametrization of the DM fraction should be conservative in the sense that, if NGC 6397 has a DM halo, $\rho_\chi$ should be much greater near the center of the cluster than at the half-light radius.

The population of WDs in NGC 6397 has been measured very well with deep HST imaging and the color-magnitude diagram of NGC 6397 exhibits a clear WD cooling sequence with a truncation at low luminosity and low temperature, as expected due to the finite age of the cluster \cite{Hansen} (see fig. \ref{fig:WDplot}).

As discussed in appendix A of ref. \cite{Hansen}, the truncation of the WD cooling sequence occurs at an absolute magnitude in the Hubble F814W filter $M_{814} = 15.15 \pm 0.15$. The best-fit model for the WD is a mass at truncation of roughly $M_{\mathrm{WD}} \simeq 0.6$ M$_\odot$ which corresponds to a cooling age 
$t_{\mathrm{cool}} \simeq 11.0\ \pm$ 0.5 Gyr. Using the cooling models of Bergeron 
\cite{Holberg,Kowalski,Bergeron,Fontaine,BergeronSite} and applying a correction for WD masses in the range 0.5-0.6 M$_\odot$, we find $T_{\mathrm{eff}} \simeq$ 3500-3700 K.

Demanding that the annihilation luminosity within the WD is less than the observed luminosity of the least luminous  WDs in NGC 6397 places constraints on the product of $f_{\mathrm{DM}}$ and $\sigma_{\chi \mathrm{p}}$. Consider first constraints on $f_{\mathrm{DM}}$ as a function of $\sigma_{\chi\mathrm{p}}$. These constraints are depicted in fig.~\ref{fig:rhovsigma} assuming the best-fit values of $M_\chi$ from a DM interpretation of the recent CDMS-Si \cite{CDMS-Si}, CoGeNT \cite{CoGent}, CRESST \cite{CRESST}, and DAMA \cite{Savage} results. Fig. \ref{fig:rhovsigma} shows that if such an interpretation of any of these results could be taken seriously, then it must be the case that either $f_{\mathrm{DM}} \lesssim 10^{-3}$ or the DM is not WIMP-like (for example, an annihilation rate lower than $\sim 10^{-36} \ \mathrm{cm}^3/\mathrm{s}$ would suffice to lift the constraint.  This is 10 orders of magnitude lower than the WIMP-like rate $\sim 3 \times 10^{-26}\ \mathrm{cm}^3$/s).
%
%	M v sigma figure
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{mvsigma}
\caption{Constraints on the DM fraction in NGC 6397, presented as a function of the DM-proton scattering cross-section and DM particle mass:  The black dotted curves correspond to the values of $f_{\mathrm{DM}}$ indicated by the labels on the figure (spanning a range of 3500-3700 K corresponding to the temperature of the WD at the truncation of the cooling sequence in NGC 6397).  Shaded regions denote the confidence contours for CDMS-Si (95\% purple, 99\% blue), CRESST (1-sigma light red, 2-sigma pink), CoGeNT (90\% grey, 99\% dark red), and DAMA (90\% cyan, 3-sigma light green, 5-sigma dark green).  The solid curves represent the exclusion limits for LUX (90\% dark green, 95\% light green) and SuperCDMS (90\% pink). If any of the observed scattering events are confirmed as due to DM WIMPs, then the fraction of DM in NGC 6397 must be $\lesssim 10^{-3}$.  The limits from the various experiments were obtained from ref. \cite{DMTools}.}
\label{fig:mvsigma}
\end{figure}
%
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The results shown in fig.~\ref{fig:rhovsigma} are interesting. Interpreting any of the anomalous signals in the variety of experiments that have reported events in direct detection detectors as broadly WIMP-like DM yields a constraint on the DM content of GCs that is far more restrictive than kinematic constraints. Furthermore, even models that exhibit no tension with the recent LUX and SuperCDMS constraints may yield observably large effects on WD cooling so as to either be constrained by WD cooling in GCs or to constrain $f_{\mathrm{DM}}$ within these GCs.

Fig.~\ref{fig:rhovmass} shows the corresponding constraints for $f_{\mathrm{DM}}$ as a function of $M_\chi$ assuming the best-fit values of $\sigma_{\chi\mathrm{p}}$ from a DM interpretation of the recent CDMS-Si \cite{CDMS-Si}, CoGeNT \cite{CoGent}, CRESST \cite{CRESST}, and DAMA \cite{Savage} results.  Again, these constraints are interesting:  If the DM model can be taken seriously, then we must have that $f_{\mathrm{DM}} \lesssim 10^{-3}$ or the DM is not WIMP-like.  On the other hand, DM parameter values that are not in conflict with the results of LUX and SuperCDMS can still place strong constraints on $f_{\mathrm{DM}}$.  Fig. ~\ref{fig:rhovmass} also shows that if the scattering cross-section $\sigma_{\chi\mathrm{p}}$ is of order $10^{-41}\  \mathrm{cm}^2$, then $f_{\mathrm{DM}} \lesssim 10^{-2}$ {\it independent of} $M_\chi$.

In fig.~\ref{fig:mvsigma}, we compare the confidence contours of CoGeNT, CRESST, DAMA and CDMS and the exclusion limits from LUX and SuperCDMS with curves of constant $f_{\mathrm{DM}}$.  We see that when we consider the full confidence contours, rather than assuming the best-fit values for $M_\chi$ and $\sigma_{\chi\mathrm{p}}$, the constraints on $f_\mathrm{DM}$ are only slightly better than our fiducial constraints:  The lowest allowed value for $f_\mathrm{DM}$ is $\sim 10^{-4}$, whereas the fidulcial constraint is $\approx 8.5\times 10^{-6}$, meanwhile the highest allowed value is no better than the fiducial constraint $f_\mathrm{DM} < 0.5$.
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%	Globular cluster figure
\begin{figure}[htp]
\centering
\includegraphics[width=9cm, height=11cm]{Globulars}
\caption{The galactic globular cluster distribution:  $\alpha$ is the ratio of the King model central concentration to the velocity dispersion of the cluster and $R_\mathrm{SUN}$ is the distance from the Sun to the cluster.  The black dots represent galactic GCs while the cyan diamond is for NGC 6397.  The shaded region represents a hypothetical selection criterion for GCs.  GCs in this region should be interesting candidates for application of our method if high quality observations of the WD cooling sequence are obtained.}
\label{fig:globulars}
\end{figure}
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\section{Conclusions and Discussion}

Stars accumulate DM in their cores as they orbit their host halos. If the DM is self-interacting, then this DM will annihilate, heating the core of the star. In the case of a WD, the luminosity from annihilations prevents the star from cooling beyond some minimum temperature. Because the annihilation rate is proportional to the local DM density, we can use the coolest WD in a GC to put an upper limit on the DM density in the cluster. In particular, we have seen that if the events observed in CoGeNT, DAMA, CRESST, and CDMS-Si could be interpreted broadly as WIMP-like DM, then observations of the WD cooling sequence in NGC 6397 limit $f_{\mathrm{DM}} \lesssim 10^{-3}$ for the best-fit values of $M_\chi\ \mathrm{and}\ \sigma_{\chi\mathrm{p}}$. This would be a significant improvement over existing, kinematic constraints on the DM content of GCs.  Additionally, DM WIMPs with parameters consistent with the exclusion limits from LUX and SuperCDMS could still potentially place a powerful constraint on $f_{\mathrm{DM}}$.

This type of WIMP astronomy may have utility far beyond what we point out here.  Note that the capture rate of WIMPs in a star is directly proportional to the local density of DM and inversely proportional to the velocity dispersion of the DM distribution.  Therefore, the most interesting environments for applying our method have the highest values of the ratio of density to velocity dispersion.  Let us define this ratio as 
%
%	Definition of alpha, the globular cluster ratio of King concentration to velocity dispersion
\begin{equation}
	\alpha = \frac{c}{\bar{v}}
\end{equation}
%
%
where $c$ is the central concentration of the King model for the globular cluster (which can be thought of as a proxy for the central density).

In order to constrain $f_{\mathrm{DM}}$ in a given environment, our method requires the observation of a cool WD.  Therefore, we should consider nearby GCs such that one could observe the full WD cooling sequence in the cluster.  Fig. \ref{fig:globulars} shows the distribution of Galactic GCs.  If we consider GCs with $\alpha > 0.5$ located within 12 kpc of the Sun, then we have 7 candidates for further observation in addition to NGC 6397.  The best target for our method is NGC 6558, which is located at a distance of 7.4 kpc from the Sun and has $\alpha = 0.81$ (for comparison NGC 6397 has $\alpha = 0.56$).


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\acknowledgments
We would like to thank Brad Hansen and Jason Kalirai for their help in locating and interpreting the observational data for NGC 6397.
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